{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Characterization of Systems in the Time Domain\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analysis of a Passive Electrical Network\n",
    "\n",
    "[Electrical networks](https://en.wikipedia.org/wiki/Electrical_network) composed of linear passive elements, like resistors, capacitors and inductors can be described by linear ordinary differential equations (ODEs) with constant coefficients. Hence, in view of the theory of signals and systems they can be interpreted as linear time-invariant (LTI) systems. The different ways to characterize the properties of an LTI system introduced before are illustrated at the example of a second-order analog [low-pass filter](https://en.wikipedia.org/wiki/Low-pass_filter).\n",
    "\n",
    "![Circuit of a 2nd-order analog low-pass filter](lowpass.png)\n",
    "\n",
    "It is assumed that no energy is stored in the capacitor and inductor for $t<0$. It is furthermore assumed that $x(t) = 0$ for $t<0$. Hence $y(t) = 0$ and $\\frac{d y(t)}{dt} = 0$ for $t<0$. For illustration, the normalized values $L = 0.5$, $R = 1$, $C = 0.4$ are used for the elements of the electrical network."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Differential Equation\n",
    "\n",
    "The differential equation describing the input/output relation of the electrical network is derived by applying [Kirchhoff's circuit laws](https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws) to the network. This results in the following ODE\n",
    "\n",
    "\\begin{equation}\n",
    "C L \\frac{d^2 y(t)}{dt^2} + C R \\frac{d y(t)}{dt} + y(t) = x(t)\n",
    "\\end{equation}\n",
    "\n",
    "This ODE is defined in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$C L \\frac{d^{2}}{d t^{2}} y{\\left (t \\right )} + C R \\frac{d}{d t} y{\\left (t \\right )} + y{\\left (t \\right )} = x{\\left (t \\right )}$$"
      ],
      "text/plain": [
       "      2                                   \n",
       "     d              d                     \n",
       "C⋅L⋅───(y(t)) + C⋅R⋅──(y(t)) + y(t) = x(t)\n",
       "      2             dt                    \n",
       "    dt                                    "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "%matplotlib inline\n",
    "sym.init_printing()\n",
    "\n",
    "t, L, R, C = sym.symbols('t L R C', real=True)\n",
    "x = sym.Function('x')(t)\n",
    "y = sym.Function('y')(t)\n",
    "\n",
    "ode = sym.Eq(L*C*y.diff(t, 2) + R*C*y.diff(t) + y, x)\n",
    "ode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The normalized values of the network elements are stored in a dictionary for later substitution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ C : \\frac{2}{5}, \\quad L : \\frac{1}{2}, \\quad R : 1\\right \\}$$"
      ],
      "text/plain": [
       "{C: 2/5, L: 1/2, R: 1}"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RLC = {R: 1, L: sym.Rational('.5'), C: sym.Rational('.4')}\n",
    "RLC"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Impulse Response\n",
    "\n",
    "The passive electrical network and the ODE describing its input/output relation can be interpreted as an LTI system. Hence, the system can be characterized by its [impulse response](impulse_response.ipynb) $h(t)$ which is defined as the output of the system for a Dirac Delta impulse $x(t) = \\delta(t)$ at the input. For the given system, the impulse response is calculated by explicit solution of the ODE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA8AAAABdCAMAAABdGP+pAAAAPFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMAELvv3c2ZVESJZnYiqzKBxeNgQeo2UQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAEsdJREFUeAHtXYm2q6gSRTOo0aTfa///XxspZpBJSMBj1rr3KJPsQgSKXQVC1++SgCqBTr0Nu/tMa/4fSqpKWIV5qpRnfPJjXdFX0HLY18VZJTC+UpD1n5RcvjzL5EtxOD4JbrNoD4vrKqB2CXTvlBp++pRc/jzz6E9zKEUS3GbRHhLVlbkJCbxTppToXaqj9UnVCZd0Etxm0YbL5UrZqASGpEnr8jDgDosRBAHDPHk7wMzzLkkTAp7dd5EE10S7CxZVhdYnjSu+fQk8kka8pzEAv4wQJpsb1tZgjY379+TR/d6HgKc4cpEE10C7DxZVhfaIpK68TUjgI3pORH07NgDP8/watiLY0pIHoGmd5+mJPw+4Q37wa239TesEyrCRTwRIcdbEGQKT4DK0HBsDi3hInWgzCOwqonIJeCe31vo/BxK8kNFy2HozDFJSABq34BkmxL05PMNad7yz8sXQeE+aE7By3H+T4AJaCRsdkaWQOtG6ZXHFnkECe0OjE1sH3a67k9Gz2wZPEiIHoHkL7sni9mluOb2gm858AjDBNwHneSZtazkrzCNT4AJaGZsJv060HPZ1cVYJDEm7QRN0MTq8brIBNY8UgND7ObygV04d4hNkKshhgQ7c884qZs5DOTVWElxAK2GjOi0ppE60Z31pL1xCApPes0SUeTWwiTCd98LwStKN5EMgBSB069By3xRSD8xiojlYmZ8RQQe+cY0VlLAlWLwaL1ZM9N8YuBpaCRutqhRSJ9po8VwZmpNAzyeu/qrPN6q6mummz5sOxDgrjJpSAOq22apS/PKE34g6nJF0YOirpDiJK7HyXu2vVFwKpT7urDpaCRudIkghdaJ1A7xizyABMQJ60SzdtMJalg2nH9Khhy0Qup8UgMhs9TZ2No3UgrW364S7KUk0k1LFFBrdI74q3morCcLhGmglbPRbI4XUiVaBnn5ja0FvaY2Qx1vH5t+flVuqW8nq9MVZF+Nzfs0wXEKnFgHD44H75XPiaeWC8K4T6cA40Tz3kJUN6zidMjdV8x27i4KroxXYQGOHkAipEu0xWfHcSdxx1AZ5vHlsY5xW9kkmt7B65Q0MF4fpF1IBE1dNaw85ehsHdx+tVNfEKkklFEObWDUtG9/z1sLdt9KCyJ0wMjYvVb59bGQLN1yGy4q7lp2MKIgY4cXJKTtJHa6od+VER6/j4O6jPQoWfQXtUWlB/iTueCNU+faxvahaKrSt+7VDgnGh5Hoe0zzhrSb+myNrxTP6LiLh7qM9CBZ9Ba1PGkHx9s+1L6tJHke77PHfkcdzYcsEDSWYAczSuCcahZMgRRC9GtdZUjap0fJLqcYE3Cnk4sh+FlA8TWKFm4T2EFj0HbThcnGk3PlcO3JsUQZ5XIWs5MarOL9uQiyqpMWHUkzCTSZsSmsq1YiEhiKw0c7+z/+UB9IbmQSpxT9uj92RVhpDtVz+WyXv8H9/hrgULrgKWnVb3IFWqXBcZfALK2fIj1Yu/eB1EnccGeTxKqnymbDxhbRJjMddJcYKAO0OjmYrUk2yvQNLJEg95wCKaD047/2/2TuwC66CVnzpN0iNos3YGknccWSQx6ukymfC5ibGR1kBUHZTQPvNd5CxdU4pkSDNogi3ygzOGjKwjeZcpTrhCsonfpzagVGTaHNJbSsnbpeCPtkkj1dJlc+EDV5WmSovEePjrAAizABoF7F3YE6ClF+FAl7djCLp87J3YLZxa4WroGUd2KhagYBiaOVmO3adxB1HBnm8Sqp8JmxOYjxWlaiLMjyrc1gBUEJjQJvxRQpTYgmaI16hbZ+mjXQoBwYUqiYRKwI1POAuewfW4crAOFo0YabnHf9jjO+AmvIkNaHllTp+EcMdR/vkcVQjVT4TNhcxPtYKINwMAK+WCY/ZOiS5SJCh74RsKhuah6fL3oFdcFW0bATmdQm6qAttUJXDEkVwx5GDPF4lVT4TthBivBgwPFYAKNQMYHm+iDLZ1oE9tMCglldWBEE55ETZO7ADroY2qQNXhlYW5cHrcO44cpHHq6TKZ8IWR4z3WAFEmwHYtdAHWx1nP8alyq+Fpoj8cJM6cKVojzejf39WfoaDPE6VEDWRx7Nho+qkUGxOK4BoMwDbCCy3Ser1MXOE7xI5FIy6ykGJ3LupFO1edcPD47jjaJ88HkNRsFdP4jjkIY9nwybVzF51X6hcQCy2Ul1FNpX11d+ML/VZQYXgVorWFGxsSBx3HKtg8ATGzk88yh7PTx7Phi0ntOipK2EhxTZrQHrZVDYguZakWAcuBLdStJpUE25jP3j75HF0kD0uU1fzUOXzYcsIDcVi++SmTLC3RKwIWEjE36mUU6xScOtEGyHxnaTWL+k+dxy1RJW3YhPegQ2JOLDJXxcjny9AZVLHflYKup/yVdwR/0xaiToKZFFVwi2GlqFO+BvMHdfKdpDHVQK4ls93m5M87sKmeAfWOBgObEr1fFi0eDVvLDG+w+aB9f1iduiial8l3GJoo0SjJg7mjqvZmiCPu7BJJEiMTN2SqJMYfwMvV1o7/Pg2YocusqY1wi2HNlI4Ink4d1zkgav6yeNObKoZgNqB6yTG1/j5j9uh018h532FcAuidYrCGemiyivccV5KAa64USR92EGejwub8IW8PYt1YKMiBQJSsYmTEWgJFfwppWrC0CqEWxBtcluGcccFQTDhQT8jj+vYFMZ/e8T4QhsrCS0qsrzYl08E5bqqEG5BtMlSC+eOJz7ih+RxFzbNO3Dae/hlbElGkYnNFpitjEdSeHh9cEuiDRS4kSycO25kDQr4JXncgQ21SIwXZ4oFif4LiejJaWWeVB3comiPytDPHU97Qg3k8QBsSSPwt7GVcuCb1rJbrlepXeCt8OrgFkW7IT7ys5MdjpQIeWsgjwdgS3oRv45t30nd8YZKKqFshcqWHg+4tvooCGJ5QUpmx00N5PHTYEvzzedonYNRvvoIp5jiKuKRvuIjisqR1FcdgVFc5XhuWBmldH41kMfPg+0gFzvsVQhP1bu5YbM4+GzcOXfJ/ay64JZG65aFL7bYFlcF5PHzYJMttXwtWj7+JToofdg8vQVdTDFWkzpzeMWqglscbbhcbCmr5I7jE/NsdY0NOxG2tJEsVmBh6RdD89e/kNDUsl34Dih7Sa71K4L7BbRhYt9JVSV3XD10eqfm/uCmsRWgguUvEtpg2vxm0nOJMbkNjhTHpHkSG0iCyF+3/CXmQwsl5fm/Ru44ykQePzO2lNaf1knMdFMKsOb5rJuT15XNqxkP40mPP8O83N/8/gjaCrnjAQcphb0SZ8YWJgGSip8tNxbxDfDeSv2sdODlmsM7XQY9aEREhQ8l/Vtoq+SOo1zapwp58T5s3TTPz8xjFh4TwZ6miDy6ddM0D2QYxhegvhjf/dq/iQra6bujANyK0R76MO1l5l/MvQQ/CA9cN3lr1h628Y7nuJ8sKjwhHbw9CQewYWVT/t9MVr8Tc3fNBly6BMasLcdJwiXg1os2v+xJiWzNUqj4lGKzkcdbw7aQieiccnCIW85wAFsm1YL6qH5949/KRH2na2G2BEYO09BScOtEq4ot31113HFpS+Ioytaw9es8T1P+/gsHsMG2WhK1wtEQt00HDfPoLdWN1p4tgfHqeDdzIbiVot0Vw8GI6rjjGanyrWErdLIvPYCNHDE059ZDkyXwizvuoh24Y2ti5PDOXQZurWgP9tP97NVxtTNWKGNR+wKMiXFXaM28+oWa0QPYsCXlPPe5FdHQU++c3PGAKfS4jbtEGeeYQheBWy3amNckKq2PrB1VWIbEOeuTs6wM0JCnPnTeybZUczyxeBnbUDuIzV5qqjVh1dVAyP0vdjyqWZMG4SaiFWYO4sqUR2pIXdxx5CGPx6FsCxvR2HaV1dkjcGx3Rr03kIR01+j9Rh2stp/7i+4G4aahlRjhJViiVXHHkUke97xBzujGsM14G3j/hXci/VVk95zlLw4liyz9i+5ZubyYtgc3EG3X36S1SpJ5Rzf3WLv/QkvAsqrEVyH1fTLJ42ZJZ8Zmom0sZNvJln6L9CJLwWe53EX74FoBlGTeMd/mTYUwTHfr5n1+pneBEvca+QTY9qCdIFxj4TDbhhMgs0HYRcu54UnmHV1/p3Q8OA/U9ujsYWXI46Sao/QVOhu27O3w6wLls1TRoqmw5Jb8dUWzPH8H7cj31fDWOH0Q47YIjd9uDR48jWMXbjd3XERh8jipzCCtC8+GLU7aDaRW1A66RlJuyQaw+Ku4g3YSKwdO6GXcFsY23S98EuO3Z99iv4zgGPx9KUiVh2pIzX46bMGCbibhIuZLL33XRGrJZvC4K2pH+3gOrxlG5zjzju1hyyq6f6eoFAoYfyDcRAWp8iA80eznw+Z+Pc4WK1rybMgUPN26MXbwXhr+sQHXZt7xvEs/fuQyPt5XKY3flDD+2AovSB6HuotmPx823jp/4kK05KnhDsRE600UADHmHSCUO3dvAvcdtegqZfxRkjwOCESza9i6aXq/N31du9gA4V/5X7TkqRGDvupGBtIY8w4Qyk3SgOGX+zPPMKUuZPxRlDwOc4zbjcw08BxDw7Zttj03dC1i6x7S/Glb6hTYiqukSNJMG1rRkieGixDRVy0wkMaYd2A54d9dGHANZAlM+eRljD+KkscBEBLfbQ3bRk0lcmoXG4X4R/6IljwzYLDwoJpol3mHfQ0s7f0CDYt14ABSVp1iFc2uYXvhGQZ04Gax1SnxUrUSLVnqCTWU+yJD6G1C2/sZY94BlV+4EgvPbrcf7cANGn8AIGkENrAhRD507WKjEP/In7/RgUF7hWfQ27ASY97B3ndwETjQ/ss6cIPGHwyQUKsPGjbsJW77SrWLjUI88OfX69yYqh/vwH0DcMH52DSR1zbKvIPKsntiS4Ynd83CbKrbM/6geORm17AtlOvTLLaY19+a9sP3D63RdQXKLZlUs6bQAsJdg4dgAbAOHJyhtoT7zb7gt/ej031qq37Z+mRzAVi2mlD6fksGPr0ptIBp1+AhEDJfAwenry7hLgV+6T+fT24/ytXBd1bI5fcr5MvGt9KdT7FFhpSu59ttST3hzn1baCmIHYOHHYhG8DI9VodbBCN9SwE3siBqqcbZ6yqdC6iXLTmC0KPEPd9KF0GBV78wI28S7Y7BQ6CYr2SnlsDC/KabXg6EIwgtbsaKoP7d9+ToGb6V7hYTs4qZ3+v6oB7igz4Q7mIjYzla7LBHc1hRM1q7wUMk9iv5KSXwpqpK08sBcwSBj88xvDushKX72egvfCvdLZ47V5VJpjJJZ4e6n+OOZWhNSGdE65bFFXsGCdAhyeblgPnCsMR9KFOgpweiSNarO0KZeQf+SM5gNfXMTt58wWwAtkA6Idp8crtKqlYCT3BBa/NywBxBWOLgPCNs6QYdGLbSXRg/rwcbgedtzs1+3HEECyj7l6JFFkgnRFtWllfpNUiAThxtXg6YIwhb3APY8h+YSbOtdAegCfEOTPNCYmbr6siaMYpNky2QTog2o+CuomqVABz+ZPVyAI4g7N4daMeFaXHAVvq8iA68Mq3ZJhPn2aHZhUaPurLBPSHa7OK7CqxOAh3YiVq9HNDB0RY3rj0+hbindtz+rfRNicpG4FFaArvPDs0uLYoW2SCdD2128V0F1iKBDkxUcHVm8D6leTmAelJHELY4cphv9wYSasBW+kbJZx14kpfAjGVfUjIGWmwuq/iJIg8/C9qSkrzKrkMCnxv3GU59n2leDqCa1I7cFvcAjY8ylkImw8fAFkw807EOrCyBXWeHQoGH/zfR6k4dyCOS0GKX65pLhV+jPSyuq4DqJTAMPXPYwE5h0bwcAAT6StviwPtBF+gEYSE2NawDsyUw+Jss7rXYgtZ0WIEBnwNt9S/fVcEcEhjBohK7XqGlaV4OIJQ6grDE0fzDykdyZ61GfETP+73e3ttMemS25rCr9AUzGQMtskBCZ0HrbIkr8iQSuEPPfTEvJBYvB9wRhCXuCT7TiC7IXE3uyOgGPXai60+6+nacHbpTTnywjhYr1pl5ODN4PxHaePlcOZqTwAsUSez4HGxupns5wJDYDo8e133osviF+8EokTLcYoCdJzZ9najfxG8YyRhobXBPg9bdClfsOSRAepNg7+OzMVQPDhtK6gjCiJu2H9FeP3s6jvqF8sSuC/u5e+M/eDq9/Q95XGeH+ksNTGGgtcA9D9pAoVzJWpYAWQX6LAk0RxAl8H7n7NC/hbZEO11lViaBBeufRp8K6guWBsyCoKx4/hbasrK8Sq9DAngnaTt8x/1THEG4k6bF6meHppXiz/W30PrlcaVoXgIjXop6QSiOILypExLoZ4cmFBGU5W+hDRLJlahxCdypJa8ThuQIwpkuMdI4OzSxHH+2v4XWL48rRfMSePkH4OYxCgCNof0P8JE6A73MN7UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$h{\\left (t \\right )} = C_{1} e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} + C_{2} e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} - \\frac{e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}} + \\frac{e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}$$"
      ],
      "text/plain": [
       "                                                                              \n",
       "             ⎛        ________________⎞         ⎛        ________________⎞    \n",
       "             ⎜       ╱   ⎛   2      ⎞ ⎟         ⎜       ╱   ⎛   2      ⎞ ⎟    \n",
       "             ⎜     ╲╱  C⋅⎝C⋅R  - 4⋅L⎠ ⎟         ⎜     ╲╱  C⋅⎝C⋅R  - 4⋅L⎠ ⎟    \n",
       "           t⋅⎜-R - ───────────────────⎟       t⋅⎜-R + ───────────────────⎟    \n",
       "             ⎝              C         ⎠         ⎝              C         ⎠    \n",
       "           ────────────────────────────       ────────────────────────────    \n",
       "                       2⋅L                                2⋅L                ℯ\n",
       "h(t) = C₁⋅ℯ                             + C₂⋅ℯ                             - ─\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "  ⎛        ________________⎞                   ⎛        ________________⎞     \n",
       "  ⎜       ╱   ⎛   2      ⎞ ⎟                   ⎜       ╱   ⎛   2      ⎞ ⎟     \n",
       "  ⎜     ╲╱  C⋅⎝C⋅R  - 4⋅L⎠ ⎟                   ⎜     ╲╱  C⋅⎝C⋅R  - 4⋅L⎠ ⎟     \n",
       "t⋅⎜-R - ───────────────────⎟                 t⋅⎜-R + ───────────────────⎟     \n",
       "  ⎝              C         ⎠                   ⎝              C         ⎠     \n",
       "────────────────────────────                 ────────────────────────────     \n",
       "            2⋅L                                          2⋅L                  \n",
       "                            ⋅Heaviside(t)   ℯ                            ⋅Heav\n",
       "───────────────────────────────────────── + ──────────────────────────────────\n",
       "             ________________                             ________________    \n",
       "            ╱   ⎛   2      ⎞                             ╱   ⎛   2      ⎞     \n",
       "          ╲╱  C⋅⎝C⋅R  - 4⋅L⎠                           ╲╱  C⋅⎝C⋅R  - 4⋅L⎠     \n",
       "\n",
       "        \n",
       "        \n",
       "        \n",
       "        \n",
       "        \n",
       "        \n",
       "        \n",
       "iside(t)\n",
       "────────\n",
       "        \n",
       "        \n",
       "        "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution_h = sym.dsolve(\n",
    "    ode.subs(x, sym.DiracDelta(t)).subs(y, sym.Function('h')(t)))\n",
    "solution_h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The integration constants $C_1$ and $C_2$ have to be determined from the initial conditions $y(t) = 0$ and $\\frac{d y(t)}{dt} = 0$ for $t<0$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ C_{1} : 0, \\quad C_{2} : 0\\right \\}$$"
      ],
      "text/plain": [
       "{C₁: 0, C₂: 0}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integration_constants = sym.solve((solution_h.rhs.limit(\n",
    "    t, 0, '-'), solution_h.rhs.diff(t).limit(t, 0, '-')), ['C1', 'C2'])\n",
    "integration_constants"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitution of the values for the integration constants $C_1$ and $C_2$ into the result from above yields the impulse response of the low-pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$h{\\left (t \\right )} = - \\frac{e^{\\frac{t \\left(- R - \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}} + \\frac{e^{\\frac{t \\left(- R + \\frac{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}{C}\\right)}{2 L}} \\theta\\left(t\\right)}{\\sqrt{C \\left(C R^{2} - 4 L\\right)}}$$"
      ],
      "text/plain": [
       "            ⎛        ________________⎞                   ⎛        ____________\n",
       "            ⎜       ╱   ⎛   2      ⎞ ⎟                   ⎜       ╱   ⎛   2    \n",
       "            ⎜     ╲╱  C⋅⎝C⋅R  - 4⋅L⎠ ⎟                   ⎜     ╲╱  C⋅⎝C⋅R  - 4\n",
       "          t⋅⎜-R - ───────────────────⎟                 t⋅⎜-R + ───────────────\n",
       "            ⎝              C         ⎠                   ⎝              C     \n",
       "          ────────────────────────────                 ───────────────────────\n",
       "                      2⋅L                                          2⋅L        \n",
       "         ℯ                            ⋅Heaviside(t)   ℯ                       \n",
       "h(t) = - ────────────────────────────────────────── + ────────────────────────\n",
       "                       ________________                             __________\n",
       "                      ╱   ⎛   2      ⎞                             ╱   ⎛   2  \n",
       "                    ╲╱  C⋅⎝C⋅R  - 4⋅L⎠                           ╲╱  C⋅⎝C⋅R  -\n",
       "\n",
       "____⎞             \n",
       "  ⎞ ⎟             \n",
       "⋅L⎠ ⎟             \n",
       "────⎟             \n",
       "    ⎠             \n",
       "─────             \n",
       "                  \n",
       "     ⋅Heaviside(t)\n",
       "──────────────────\n",
       "______            \n",
       "    ⎞             \n",
       " 4⋅L⎠             "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = solution_h.subs(integration_constants)\n",
    "h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The impulse response is plotted for the values of $R$, $L$ and $C$ given above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 52.160938 270.246469 \n",
       "L 52.160938 28.326469 \n",
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       "   </g>\n",
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       "    <path d=\"M 376.815483 282.342469 \n",
       "L 376.815483 16.230469 \n",
       "\" style=\"fill:none;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 19.695483 228.629262 \n",
       "L 376.815483 228.629262 \n",
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       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 19.695483 16.230469 \n",
       "L 376.815483 16.230469 \n",
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       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pfaa542fe45\">\n",
       "   <rect height=\"266.112\" width=\"357.12\" x=\"19.695483\" y=\"16.230469\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(h.rhs.subs(RLC), (t,-1,10), ylabel=r'h(t)');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step Response\n",
    "\n",
    "The [step response](step_response.ipynb) is derived by integrating over the impulse response $h(t)$. For ease of illustration this is performed for the specific values of the elements given above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{5 i \\left(- \\frac{e^{t \\left(-1 - 2 i\\right)}}{5} + \\frac{2 i e^{t \\left(-1 - 2 i\\right)}}{5} + \\frac{1}{5} - \\frac{2 i}{5}\\right) \\theta\\left(t\\right)}{4} - \\frac{5 i \\left(- \\frac{e^{t \\left(-1 + 2 i\\right)}}{5} - \\frac{2 i e^{t \\left(-1 + 2 i\\right)}}{5} + \\frac{1}{5} + \\frac{2 i}{5}\\right) \\theta\\left(t\\right)}{4}$$"
      ],
      "text/plain": [
       "    ⎛   t⋅(-1 - 2⋅ⅈ)        t⋅(-1 - 2⋅ⅈ)          ⎞                    ⎛   t⋅(\n",
       "    ⎜  ℯ               2⋅ⅈ⋅ℯ               1   2⋅ⅈ⎟                    ⎜  ℯ   \n",
       "5⋅ⅈ⋅⎜- ───────────── + ───────────────── + ─ - ───⎟⋅Heaviside(t)   5⋅ⅈ⋅⎜- ────\n",
       "    ⎝        5                 5           5    5 ⎠                    ⎝      \n",
       "──────────────────────────────────────────────────────────────── - ───────────\n",
       "                               4                                              \n",
       "\n",
       "-1 + 2⋅ⅈ)        t⋅(-1 + 2⋅ⅈ)          ⎞             \n",
       "            2⋅ⅈ⋅ℯ               1   2⋅ⅈ⎟             \n",
       "───────── - ───────────────── + ─ + ───⎟⋅Heaviside(t)\n",
       "  5                 5           5    5 ⎠             \n",
       "─────────────────────────────────────────────────────\n",
       "                    4                                "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau = sym.symbols('tau', real=True)\n",
    "\n",
    "he = sym.integrate(h.rhs.subs(RLC).subs(t, tau), (tau, 0, t))\n",
    "he"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the step response"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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      ],
      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(he, (t,-1,10), ylabel=r'$h_\\epsilon(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transfer Function\n",
    "\n",
    "For an exponential input signal $x(t) = e^{s t}$, the [transfer function](eigenfunctions.ipynb#Transfer-Function) $H(s)$ represents the complex weight of the exponential output signal $y(t) = H(s) \\cdot e^{s t}$. The transfer function is derived by introducing $x(t)$ and $y(t)$ into the ODE and solving for $H(s)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{C L s^{2} + C R s + 1}$$"
      ],
      "text/plain": [
       "        1         \n",
       "──────────────────\n",
       "     2            \n",
       "C⋅L⋅s  + C⋅R⋅s + 1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = sym.symbols('s')\n",
    "H = sym.Function('H')(s)\n",
    "\n",
    "H, = sym.solve(ode.subs(x, sym.exp(s*t)).subs(y, H*sym.exp(s*t)).doit(), H)\n",
    "H"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transfer characteristic of an LTI system for harmonic exponential signals $e^{j \\omega t} = \\cos(\\omega t) + j \\sin(\\omega t)$ is of special interest for the analysis of electrical circuits. It can be derived from $H(s)$ by substituting the complex frequency $s$ by $s = j \\omega$. The resulting transfer function $H(j \\omega)$ provides the attenuation/amplification and phase the system adds to an harmonic input signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{- C L \\omega^{2} + i C R \\omega + 1}$$"
      ],
      "text/plain": [
       "          1           \n",
       "──────────────────────\n",
       "       2              \n",
       "- C⋅L⋅ω  + ⅈ⋅C⋅R⋅ω + 1"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = sym.symbols('omega', real=True)\n",
    "\n",
    "Hjw = H.subs(s, sym.I * w)\n",
    "Hjw"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The magnitude of the transfer function $|H(j \\omega)|$ is plotted for illustration for the specific values of the elements given above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(abs(Hjw.subs(RLC)), (w, -10, 10),\n",
    "         ylabel=r'$|H(j \\omega)|$', xlabel=r'$\\omega$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It now becomes obvious, that the low frequencies pass through the system and that high frequencies are attenuated. This motivates the term 'low-pass' for such systems.\n",
    "\n",
    "As alternative to the solution of the ODE, the transfer function $H(s)$ is [computed from the impulse response](eigenfunctions.ipynb#Link-between-Transfer-Function-and-Impulse-Response) and plotted for the specific values of the elements given above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{5}{s^{2} + 2 s + 5}$$"
      ],
      "text/plain": [
       "     5      \n",
       "────────────\n",
       " 2          \n",
       "s  + 2⋅s + 5"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H2 = sym.integrate(h.rhs.subs(RLC)*sym.exp(-s*t), (t, 0, sym.oo), conds='none')\n",
    "H2.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(abs(H2.subs(s, sym.I*w)), (w, -10, 10),\n",
    "         ylabel=r'$|H(j \\omega)|$', xlabel=r'$\\omega$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The phase of the transfer function $\\varphi(j \\omega) = \\arg \\{ H(j \\omega) \\}$ provides insight into the phase added to an harmonic signal when passing through the system. It is computed and plotted for the specific values of the elements given above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "phi = sym.arg(Hjw)\n",
    "sym.plot(phi.subs(RLC), (w, -10, 10),\n",
    "         ylabel=r'$\\varphi(j \\omega)$', xlabel=r'$\\omega$');"
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
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